Anomalous birefringence of swollen lamellar phases : blue smectics

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Anomalous birefringence of swollen lamellar phases : blue smectics

F. Nallet, Ph. Barois

To cite this version:

F. Nallet, Ph. Barois. Anomalous birefringence of swollen lamellar phases : blue smectics. Journal de

Physique II, EDP Sciences, 1994, 4 (6), pp.1049-1060. �10.1051/jp2:1994183�. �jpa-00248013�

Classification Physic-s Abstracts

42,10Q 61.30 82.70

Anomalous birefringence of swollen lamellar phases

_:

blue smectics

F. Nallet and Ph. Barois

Centre de recherche Paul-Pascal, CNRS, Avenue A.-Schweitzer, 33600Pessac, France

(Received 3 December 1993, receii>ed in final form 10 February 1994, accepted 18 February 1994)

R4sum4. La

birdfringence

d'une

phase

lamellaire

lyotrope

est calculde _en fonction de la dilution.

Nous _montrons qu'elle _peut s'annuler h condition que la

birdfringence

naturelle des bicouches de

tensioactif soit

positive.

La dispersion _au

voisinage

du

point

de

birdfringence

nulle est calculde.

Ces rdsultats _sont confirmds par des _mesures de

spectrophotomdtrie:

la transmission des dchantillons dtudids _entre

polariseurs

croisds s'annule exactement pour une

longueur

d'onde particulibre. La variation de l'intensitd transmise _en fonction de la

longueur

d'onde _et de

l'dpaisseur

des dchantillons _esi conforrne _aux

prdvisions

du moddle.

Abstract. The

birefringence

of _a

lyotropic

lamellar phase is calculated _{as a} function of dilution.

It is found to vanish and

change sign, provided

the natural

birefringence

of surfactant bilayers is

positive. Dispersion

is calculated about the

point

of _zero

birefringence.

These

predictions

are

illustrated with experiments of

spectrophotometry:

the intensity of light transmitted between crossed

polarizers

through several lamellar samples vanishes _as expected at some

particular wavelength.

The

dependence

of the transmitted

light

_on wavelength and cell thickness is consistent with

theory.

1. Introduction.

It has been known for decades in chemical

[I]

and

biological [2]

_systems that

amphiphilic

molecules _can self-assemble to form

supramolecular

_aggregates such _as

micelles,

rods,

lamellae _or

complex

bidimensional

periodic

networks of

non-intersecting

surfaces.

The

macroscopic

_appearance of such

lyotropic phases

is then related _to the

long-range organization

of these aggregates a disordered solution of micelles for instance _can flow like a

liquid

whereas _a cubic

phase

cannot.

Liquid-crystalline

structures

(hexagonal,

lamellar I-e- smectic A _or

nematic)

_are characterized

by

their

optical birefringence.

In the

particular

_case of

lyotropic

lamellar

phases,

the uniaxial symmetry ^{is reflected}

by

_any

second rank tensor that _must be

anisotropic

with

only

two different

eigenvalues.

This _structure for the dielectric _tensor leads _{to an}

optical

axis

perpendicular

to the average

plane

^{of the}

layers

with _a

birefringence

(I.e,

extraordinary

index _n~ minus

ordinary

index

n~) usually positive [3].

1050 JOURNAL DE

PHYSIQUE

II N° 6

Highly

diluted

lyotropic

lamellar

phases

have

recently

received considerable theoretical

[4- 6]

and

experimental

attention

[7-9].

^It has been observed in several _cases that the

birefringence

exhibits _a

peculiar

behaviour

[10,

1] : for _some

particular

value of the

dilution,

_a non-oriented

sample

would appear

uniformly

dark and

slightly

coloured

(actually

blue in _most of the

reported experiments)

whereas at

higher

and lower volume fractions of the

solvent,

the usual

bright

colours of _a

strongly birefringent

material _are observed. On the other

hand, X-ray

and neutron diffraction pattems

[12]

_are

perfectly regular

when

passing through

this

particular

dilution

(I,e. they

^show

Bragg

reflections at a wave vector qo =

2

«Id,

the

layer spacing

d

increasing continuously

_upon

dilution).

The aim of this paper is to show that this anomalous behaviour _can be

simply

understood _as the

vanishing

and the

change

of

sign

of the

birefringence

of the lamellar _structure for _a

particular

dilution. Such _a

change

^of

sign

^{of the}

birefringence

had been

reported long

_ago in aqueous solutions of AOT

(sodium sulpho-di(2-ethylhexyl)

succinic

ester) [10].

We claim in the present _paper that

simple geometrical

effects

(so-called form birefringence)

can

explain

this

uncommon behaviour.

The

birefringence

of _a

regular

stack of

birefringent

lamellae swollen

by

an

isotropic

solvent is calculated in the next section. It is found _to vanish for _some

particular

value of the

layer

spacing.

The effect of the thermal fluctuations of the membranes is also considered. In section

3,

the

dispersion

of the material is calculated about the

wavelength

for which the

birefringence

vanishes. Section 4 is then devoted _{to an}

experimental

illustration of these results the transmission of visible

light through samples

of different thicknesses between

crossed

polarizers

is measured _{as a} function of the

wavelength.

2.

Birefringence

of _a

regular

stack of lamellae in _an

isotropic

solvent.

The form

birefringence

of _a

regular assembly

of thin

parallel plates

of thickness ej and dielectric _{constant sj} in _a continuous medium

(solvent)

of thickness e~ and dielectric constant s~ has been known for _a

long

time. In _terms of refractive

indices,

it may be written _as

[13]

~2 ²

fi f2 (n( ni)~

~ ~°

f

~2 _~

~ ^~2 (~.l)

2 2

where

fj

=

ej/(ej

+

e2)

and

fi

" I

fj

are the volume fractions of the

plates

and of the

solvent

respectively

and

n)

= s,, ^I = 1, ^2.

Note that the

assembly always

behaves like _a

negative

uniaxial

crystal, regardless

of the

sign

of the difference

n( n(.

The

birefringence

of _a real

lyotropic

lamellar

phase

is

slightly

_more

complex

for _two

reasons :

I)

the diluted membrane is

usually

constituted of _a sheet of

parallel

molecules of

surfactant and therefore

birefringent

in itself and

it)

thermal fluctuations of the membranes will affect their

shape

and therefore the

birefringence

^of the structure. If the intrinsic

birefringence

of the membranes is

positive,

the

birefringence

of the lamellar structure is

expected

to be also

positive

at low solvent content but _can become

negative

at

high swelling

if the form

birefringence (2,I) prevails.

Indeed,

at

high swelling

in the _case of lamellar

phases

stabilized

by entropic repulsion only [4],

the membranes are

crumpled [6-9]

and their intrinsic

birefringence

falls off faster with

dilution than the form

birefringence

of the stack. This is shown in the

following,

where the

birefringence

of the structure is calculated in _a unit cell of thickness d and

projected

area A

(from

_{now on,} ^{d refers} to the

period

of the lamellar

stacking

and _e to the thickness of the

membranes).

The electric E and

displacement

D fields in the solvent and in the membrane _are

respectively

~~'

_{~° ~~~~~}

(2.2)

DMI _{~ So EMU}

EMj

where Es is the dielectric constant of the solvent

(assumed

_to be _an

isotropic liquid)

and

s~,~

^{the dielectric tensor of the membrane}

(s~

is the

permittivity

of free

space).

At this stage, ^{it is} convenient to define the local trihedron

(I,

_m,

n)

where _n and

I, m are unit _vectors

perpendicular

and

tangent

to the membrane

respectively

at

position

r ₌

(x,

_y

(see Fig.

I

).

Primed and

unprimed

letters denote _{vector or tensor} coordinates in this trihedron and in the

macroscopic

reference frame

(x,

_{y, z}

=

optic axis) respectively.

z

n e

m

'

y

~f

' ^'

'

X ^'

'

i

Fig,

I. Definition of the local trihedron of unit _vectors (I, m, n) ^and of the angles ~ and 6. The reference frame (x, _{y, z} is fixed and _z is

along

the

optical

axis. The normal to the membrane at

position

_r

is _n, and lies in the _n-z plane with its

projection

onto rite x-y plane (dashed line)

defining

the

angle

~.

The local

relationship

between the Ds~ ^and

D~~

^is easy ^to express in

(I,

_m,

n)

since the normal component ^{of the D-field and the}

tangential

component of the E-field _are continuous _at the membrane-solvent interface

The effective

macroscopic

dielectric tensor s~~~ is defined _as

jj efff _{(~ ~)}

, " E0 ~<j

j

in which the bars denote

spatial

_{averages over} ^the unit cell of volume dJA

I,

=

~ j _d~R

Xs,

+

d~R ~,j

(2.5)

dA

sow memb

with X

=

D _or E.

1052 JOURNAL DE PHYSIQUE II N° 6

If 6

(x, y)

denotes the local

angle

between the norrnal

n(x, y)

and the

optical

axis _z, the differential volume element of the membrane is _e dx

dy/cos

6

(x,

_y with the

assumption

that the membrane thickness _e

(or equivalently

the _area _per

surfactant) keeps

constant as the

membranes fluctuate. The total volume of the membrane in the unit cell is then

eA

(I/cos 6)

the

(I/cos 6)

term, first introduced

by

Helfrich and Servuss

[14]

has various consequences

experimentally

observed in

lamellar,

vesicles and _Sponge

phase [15]

which support ^the constant thickness

assumption.

The average values _are then _:

xi

=

xs Ii

^~

_~~l

~ ⁺

i h

(2.61

X

=

D _or

E,

= 1,

2,

3 and the brackets

(. )

denote thermal averages ^over wavevectors

larger

than those of visible

light.

Inserting

the three relations

(2.3) expressed

in the

(x,

_{y, z}

)

frame into the six

equations (2.6)

will lead to the form

(2.4). b

_for

example

can be calculated _as :

D,

=

s~(ss (I ^I

_d

^Es,

⁺

iA,~Egj ^(2.7)

cos 6 d

M,~ _E~'i

(M~

^'

)u

With

A,k

_"

~~~

E~ ~

o o

and E~'i ⁼ o E~

~

o

o o Es

The matrix

M~~(6,

q~

) changes

coordinates from the

(x,

_y,

z)

set of _axes

(unprimedl

to the (I, _m, n

)

trihedron

(primed)

_:

X,

= M,~

Xj.

^{The matrix} _A,~ ^is

diagonal

_as

expected

after trivial

angular averaging

_over the azimuthal

angle

_q~.

Calculating ^f

likewise and

eliminating

the

Es,'s

leads to the final result for small 9

I ₊ ^~

l~~~

^~~

~eft

~2

~

~ ~~

(2

8a)

I o S

~ ~

] ₊ ^{~ ~ ~~}

(9~)

d 2 _EM

I

' ₊ ^~

~'~~

~~

l19~)

~~~~ ~~ ~~

~

2

~~'~~~

j _~ ^{~ ~S} ~mll

j ^{IS )}

~ _~M

I ~

At lowest

(zeroth)

order in

9~,

the

birefringence

is therefore controlled

by

:

~2

~2

^{e (ES EMI}

)(EM

I ES

)(I e/d)

_{+ E~}

(E~

j E~

~

~ ~ ~

~MI +

(e/d) (

_{ES EM}

I

~~ ~~

It vanishes for _:

~

E$(E~

~

E~j)

/

=

(2, lo)

~ (ES ~M

)(

_EM

I

~S)

As

(I e/d)

is

always positive,

the condition

(2, lo)

can be fulfilled in _{two cases}

only I) positive birefringence

of the membrane

(s~

i > s~

~

and Es > s~ i or Es < s~ _~ ;

ii) negative birefringence

of the membrane and e~ i < es < s~~.

Films of surfactant

usually

exhibit _a

positive birefringence.

Case

I)

is therefore the _more

Iikely

^observed.

Thermal averages of

angular

fluctuations of the membranes

(9

_{~) over} wavectors

larger

^than

those of visible

light obviously depend

_on the systems: lamellar

phases

stabilized

by

electrostatic interactions will fluctuate much less than

sterically

stabilized systems.

nj nj

is

plotted

as a function of the lattice

period

d in

figure

2 for _{a set} of flat

rigid

membranes

(2.9)

and

undulating

flexible membranes

(2.8).

In both _cases, the

birefringence

is

positive

at

small lattice

period d,

it falls off with

dilution,

_passes

through

_zero to reach _a

negative

minimum and vanishes

again asymptotically

as I/d.

o.ooi ^'

k~

₌ I-O

kBT

~o 5

io~4 _k~

= loo

kBT

d

m~

~ o

-5,

10~~

loo 200 300

Lamellar

period

d

Ill

Fig.

2. Plot of the dielectric anisotropy

n) n( against

the

period

of the lamellar phase. The relative

perrnittivities

are e~ 2.02 for the solvent (e.g. dodecane), _e~j =1.83 and e~~ =1.82 for the

membrane of thickness

20i.

_{The two}

curves correspond _to different membrane

bending

moduli

k~ : I k~ 100 k~ ^T (rigid membrane), dashed line and 2 k~ = 1.0 k~T (flexible membrane), solid line. Thermal fluctuations of the flexible membrane _are calculated assuming that the lamellar

phase

is stabilized

by

steric repulsion

only,

which

yields

_:

lo

~) (2 gr)~ ^' _kBT/k~ ⁱⁿ (dla) [16].

The

vanishing

of the

birefringence

is thus

expected

when the

negative

contribution of the dilution

(I.e.

form

birefringence)

compensates

exactly

the

positive birefringence

of the membranes.

Figure

2 shows that this condition should be met in

typical lyotropic

_systems.

The dielectric

perrnittivities

_{Es, s~}

i and s~~ however

depend

on the

frequency

of the

electromagnetic

wave within the

optical

range so that for _a

particular

dilution e/d

equation (2.10)

is satisfied for

a

single pulsation

_w~

only.

^The

optical

behaviour of such _a lamellar

phase

about this

particular

value _wo will be calculated in the _next section.

3.

Dispersion

about the dilution of _zero

birefringence.

We now consider _a

macroscopic

uniaxial lamellar

phase.

The vector field _n denotes the

optical axis, perpendicular

to the average

plane

of the

layers.

The

(frequency dependent)

dielectric

tensor s,~ reads

E,j ⁼

P6,j

+ As

in, ^{nj 3,j 13. ii}

1054 JOURNAL DE PHYSIQUE-II N° 6

The

ordinary

and

extraordinary

indices of refraction _are

given by nj

= P-1/3 As and

nj

= P ₊ 2/3 As

respectively.

As

argued

in the

previous section,

the

optical anisotropy

AE

depends

on the dilution of the lamellar

phase

and may ^vanish at some

frequency

_wo in the visible range. Since _we want to describe the

optical properties

of the lamellar

phase

in the

vicinity

of wo, ^we shall consider _a lowest order

expansion

of AE

w wo

AE

=

3Ej (3.2)

wo

We will _see that this

dispersion gives

to the

weakly-birefringent

_« blue smectic

» its

peculiar anisotropic optical properties

in the

vicinity

of _wo.

In _a coordinate system defined _as follows _: n, =

3,~,

k ₌

k(sin

9,

0,

_cos

9),

^the two

propagative

modes read _:

ii ordinary

wave

~2

~

~2 (3.3a)

o 2 °

and

polarization

such that D is

perpendicular

to both k and _{n ;}

iii extraordinary

_wave :

w

2

nj _n(

kj

=

(3.3b)

c~

nj sin~

_{9 +}

nj cos~

₉

and

polarization

such that D is

perpendicular

to k, in the

(k, n~plane.

If _a well

aligned sample

of such _a lamellar material of thickness D is sandwiched between crossed

polarizers,

the

intensity

of the transmitted

light

_can be calculated _{as a} function of

frequency

_w

(or wavelength

in _vacuo

Al

and direction 9 referred to

optical

axis of the incident

light

as

[13]

~~°' ~' "

=

sin~

₂

a

sin~

^~ _(k~

k~)j ^(3.4)

lo

2

in which

lo

^{is the}

intensity

of the incident

light linearly polarized along

direction _x and _a is the

angle

between this direction of

polarization

and the

projection

of the

optical

axis _{on a wave}

plane (Fig. 3).

The transmitted

intensity I/Io

is calculated and

plotted against wavelength

A for _a well

aligned sample

with

optical

axis oriented

perpendicularly

to the direction of

propagation

I.e.

6

= ar/2

(Fig. 4a)

and for _a

randomly

oriented

sample

I.e. _a

powder

in the

crystallographic

_sense

(Fig. 4b).

The parameters we have chosen in this illustration _{are :} P ₌ 2,

independent

of

frequency AE(w

= 10~

~(w wo)/wo

and

wavelength

Ao = 5 lo _nm.

Three thicknesses D

= 5, 20 and 40 _{mm are}

represented.

The relative transmitted

intensity I/Io

is _zero at

wavelength

A

o for both well

aligned

and

randomly

oriented

samples.

It oscillates

on both sides of A~ at a scale that

depends

on the

optical path

D. The

period

is

actually proportional

to I/D for the

aligned crystal

whereas

powder averaging strongly

decreases the

amplitudes

and

slightly

^shifts the extrema of the

secondary

oscillations. In the

experimentally

common case of _a

randomly

oriented

sample

we note that _most of the transmitted

light

has _a

shorter

wavelength

than A

o. The

sample

would then appear as blue in the

example

of

figure

4b.

However, the characteristic

wavelength

A

o for which the

birefringence

_goes to zero

depends

_on

~

X

' ~l

' '

n

, '

, '

'

"

0

~

^z

y

Fig.

3. Schematic

drawing

of the

light

transmission experiment. The _wave vector k of the

propagating light

is along axis _z. The

polarization

P of the incident

light

is

along

axis _x, with _a crossed analyzer A along axis y. The orientation of the

optical

axis _n of the

birefringent

material is described in spherical

coordinates by co-latitude e and azimuth _a.

~O

~ ~=~o

)

~

3

~

~

nu 0.5 20

j

(

I

₅

#

mm

o ^' a)

300 400 500 800 700

Wavelength (nm)

J 0.3

)

~

^{~ ~~}

i

0.2

3

'i

~ i 20

Z O-I _'

(

'

w ,

I ',

₅

#

mm

o ^' b)

300 400 500 800 700

Wavelength (nm)

Fig.

4. Relative transmitted

intensity

between crossed polarizers, calculated from relations (3.3) and (3.4), _{as a} function of the wavelength in i>acuo of the incident

light. Figure

4a

corresponds

to an oriented

material, with

optical

axis

perpendicular

to rite

propagation

direction (0

=

gr/2 and _a arm, while

figure 4b is relevant for _a powder sample. ^{Three different}

optical

paths D have been considered. The

wavelength

at which the birefringence is _{zero is} lo ₌ 5lo _nm

(for

other parameters, see text) note that

zero transmission _{occurs at} wavelength lo, whatever the optical path.

1056 JOURNAL DE PHYSIQUE II N° 6

the

layer spacing

_as shown in section 2 the transmitted colour thus varies with the

weight

fraction of surfactant. This

point

is consistent with

experimental

observations (see next

part).

At

frequency

_wo, it would appear from the constitutive

equation (2.4)

that the uniaxial medium is

optically isotropic.

This is not

quite

true, because

equation (2.4)

is

actually

the lowest order term of _an

expansion

in

spatial

derivatives of the electric field. The

general

relation between the electric field E and the dielectric

displacement

D should read

D,

= EojE,~ E~ ⁺ p,~~i

a(I

_E~ +

(3.5)

in which

p,~~i

^is a

phenomenological

tensor of rank 4 with uniaxial

symmetries.

There _{are no} first-order derivatives of the field in this

expression,

since their presence is ruled out

by

the inversion symmetry ^{of the} systems we consider. Of _course, such _terms have _to be written when there is _no inversion symmetry

they

_are

responsible

^{for the}

optical activity

of

isotropiq

solutions of chiral

compounds,

for instance

j13].

The second-order derivatives generate a tensor contribution of rank 2 p,~~i k~

ki

in which the wavevector components _k~

ki

kill the uniaxial symmetry. ^{It follows that the} cancellation of the

birefringence

_may not occur at all incidences 9. As _a result, the

powder sample

should transmit

a weak

intensity (of

order

p)

between crossed

polarizers

at

frequency

_wo. We shall _see in the next section that this effect is _not detected

experimentally

which confirms that the

p,~~i's

contribution is

negligible.

At last, _one should _note that _we used in

figure

4 the

simple

first-order

expansion

of AE

(3.2)

all _over the

frequency

_range of visible

light.

For

frequencies

_w

significantly

different

from _wo

higher

^order terms in

(w

_w

o

)/wo

_are

expected

to become

important. Figures

4a and 4b _must therefore be considered _as accurate close to wo and

only qualitatively

correct

elsewhere.

4.

Experimental

section.

An anomalous

birefringence

of lamellar

phases

similar _to that described in the

previous

sections has been

reported

in several

lyotropic

_systems

_{j10, 11].}

In order _to check the relevance of _our

description,

_we have chosen _to

investigate

the quatemary system ^Sodium

Dodecyl

Sulfate

(SDS), hexanol,

dodecane and water. This system ^forms a Iamellar

phase

_over a wide range ^{of oil} concentrations

(see Fig.

5 for _a

pseudo-temary phase diagram).

Samples prepared along

the dotted line of

figure

5 exhibit a dark

unique

colour between crossed

polarizers (from

blue to

yellow depending

on the

composition).

This feature shows

an

an°ma'Y

of the

birefringence

whereas _neutron diffraction patterns show _no

singular

behaviour of the

layered

structure

j12b].

Four different

compositions

_were

prepared

with the _same

weight

fraction of hexanol

(x~

=

16

fb)

and _constant

water/surfactant

ratio

(xw/xs

=

4.3 ).

They

_are labelled relative _to their dodecane

weight

fraction x~ 545

(x~

= 45.0 iii

), 548,

S51 and 554. After

preparation,

the mixtures were left several

days

at room temperature

(20 °C)

until

they

look clear and

homogeneous.

All the

samples

showed coloured

patches separated by

darker

regions

between crossed

polarizers.

The observed colour _was

obviously

characteristic of the

composition.

The

photometric

measurements were carried _out in the visible range

(300-700

nm) with a

Perkin-Elmer Model 330

spectrophotometer.

The

samples

_were introduced in

amorphous

quartz ^{cells with}

optical paths

of 5,

lo,

20 and 40 _mm. Two identical cells _were

prepared

in each _{case to} balance the two ways of the

spectrophotometer

_{: one was put} between crossed

polarizers

in the main beam (transmitted

intensity I~)

whereas the other _{one was} mounted between

parallel polarizers

in the reference beam

(transmitted intensity

_Iii

).

^The recorded

signal I~/Iii

was therefore insensitive to the

wavelength-dependent absorption

of the

polarizers,

the cells and the

samples.

HEXANOL

12

t

WATER,

SDS

DODECANE

Fig. 5. Pseudo-ternary

phase diagram

at temperature ^T =

21 °C of the SDS/hexanol/water/dodecane system, ^{taken from reference} ii I] (see also Ref. [17]). The water _over surfactant _mass ratio ,rw/xs is 4.3. The line shows the dilution path ^{followed in the lamellar} L~ phase ⁱⁿ our experiment.

I~ and I, _are

respectively

direct and inverse micellar phases. t is _an

I~ Iz L~ ^three phase equilibrium.

A

typical

series of spectra ^{for different} thicknesses is

reported

in

figure

6

(sample 551).

The

ratio

l~/Iii

is _zero

(I.e.

less than 4

x10~~)

for _a

particular wavelength

Ao ₌ ^393±

5 _nm. Oscillations on either sides of this fixed

point

_{Ao are} then observed _{at a} scale that gets shorter upon

increasing

the thickness of the cells.

The theoretical value of the measured ratio

l~/iii

is

easily

obtained from

equation (3.4)

If ₍

^~

^theor

^{I w , 6, a}

^)/I

^o

'

I(~°,

9, _"

)/10

^{~~ ~~}

Equation (4,

I

applies

for _a

single crystal. Angular averaging

_{over a} and 9 with

appropriate

distribution function has _to be

performed

^for our non-oriented

samples.

The theoretical ratio

I~

_Iii( is

plotted

in

figure

7 in the _case of _a

randomly

oriented

sample.

It vanishes at

wavelength

A

o and oscillates _on both sides with _a finite

amplitude

_: the maxima and

secondary

minima _{are non-zero} and lower than I. In the _case of _a well oriented

sample (not plotted),

the behaviour is

qualitatively

the _same but with different

amplitudes

the ratio

I(w,

6

=

ar/2,

«)/Io

oscillates between _zero and

sin~2«

_{which in turn}

implies

that

I~

Iii( ^{oscillates between} zero and

tan~

₂

« I.e. with unbound

amplitude.

In _our

experiments,

^{the situation} is

obviously

intermediate _our

samples

_are neither oriented

nor

perfectly isotropic

since the

colouring

between crossed

polarizers

is not uniform.

Indeed,

the

amplitude

of the oscillations _on either sides of A~ can

change significantly

_upon

ageing

_or

gentle shaking.

We have _not tried to fit the

experimental

curves since their actual

shapes

depend

too much _{on an} unknown

angular

distribution function of the

crystallites.

1058 JOURNAL DE

PHYSIQUE

II N° 6

1.5

tz~ 20 _mm

/ o C

~i

b

₄₀

il mm

11 0.5

5 10 _mm

0.0

300 400 500 600 700

Wavelength (nm)

Fig.

6. Transmitted

intensity

between crossed polarizers (1~ relative _to the _one between parallel

polarizers

(I« ), ^for _a

particular

sample ⁵⁵¹ along the dilution line, _{as a} function of the wavelength

in vacuo of the incident light. The

optical

path _ranges from 5 _mm to 40 _mm. The wavelength of _zero

transmission is about 393 _nm, independent of the

optical path.

o.5

)

_0.3

I _1'

#

°.2

,'i10

j

_, , mm

j

_{o i} ^I j

,'

_{5 mm}

£ ',

I ^,

',

_, '

0

300 400 500 600 700

Wavelength (nm)

Fig.

7. -Theoretical variation of the ratio of the transmitted

intensities1~

(crossed

polarizersi

to

ii (parallel polarizers) in the _case of _a

randomly

oriented

sample.

The

wavelength

of _zero transmission is set to 393 _{nm to} match the

experimental

value (see Fig. 6) and the _same four

optical paths

D _are

represented. Other parameters are e =

2.0 and he

=

2 _x lo-

~(w

wo)two. The

experimental

behaviour is

qualitatively

well reproduced. The amplitude of the secondary oscillations is found _to increase when the distribution of orientation of the

optical

axis changes from isotropic to

partially

oriented, which is

consistent with experimental observations (see text and

Fig.

6).

The

experimental

^behaviour is however very similar to the calculated

ratio1~ /lj (Fig. 7).

All

samples

exhibit _{a zero}

birefringence

for _a

particular wavelength

_A~ which

depends

_on

composition (see Tab-I

but not on the thickness of the

sample (Fig. 6). Secondary

oscillations

are observed in all

samples

with _a

period

that decreases upon

increasing

thickness. The value of the

secondary

minima is _non-zero and

changes slowly

_upon

ageing.

We

interpret

this

observation _{as a} slow variation of the distributions of

angles

_a and 9 with time.

Table I.

Wavelengths

_Ao

for

which the transmission between crossed

polarizers

vanishes

exactly, given for four different samples along

_a line with _constant hexanol

w>eight

content

(x~

=

0.16 and constant water over

surfactant

ratio

(>.w/xs

₌ 4.3 ). Each

sample

_was tested

in

four

cell~i with

different

thicknesses

(5, lo,

20 and 40

nm).

Variations in A

~ with thickness

were less than 5 _nm

(1.5 fb).

Weight

fractions

(iii) Wavelength

of _zero

birefringence (nm)

545

7.4/16.0/31.6/45.0

320

548

6.8/16.0/29.2/48.0

350

S51

6.2/16.0/26.8/51.0

393

554

5.7/16.0/24.4/54.0

477

At last, _{we note} that the first-order

approximation

of AE in powers of (w

w~)/wo (3.2) reproduces remarkably

well the

experimental

behaviour

higher

order terms do not seem to contribute very much in the visible range. This does _{not mean} however that the

experiments

_are

not sensitive _to the real variations of AE with

frequency.

In _terms of

wavelength, (3.2)

reads

A~

A~

=

3E~ (4.2)

which

implies

that AE varies

relatively quicker

with A below A

o than above. This is

actually

^the

case in _our

experiments

_: the

period

of the oscillations of the transmitted

light

is

clearly

shorter below A

o.

5. Conclusion.

We have shown in this paper that the anomalous

birefringence

of diluted

lyotropic

smectics _can be

simply

understood _as the

vanishing

of the

birefringence

_{at some}

particular

dilution for which the

negative

form

birefringence

of the stack of lamellae matches

exactly

^the

positive birefringence

of each lamella. Exact cancellation

only

occurs at a

particular wavelength

Ao ^which

depends

on

composition. Samples

illuminated with white

light

between crossed

polarizers

transmit all

wavelength

but A~ and thus exhibit uniform

pastel

colours.

Measurements of transmitted intensities between crossed

polarizers

_are consistent with theoretical

expectations

and hence support ^{the model.}

Close _to A~, ^the

birefringence

and

subsequently anisotropic optical properties

_{are very}

sensitive to thermal undulations of the membranes (see formulae

(2.8)).

This

phenomenon

could be used _to

probe

the effect of temperature or external fields _on these fluctuations for

instance.

Acknowledgments.

We wish to thank D. Roux and G. Porte for

helpful

discussions.

References

iii Ekwall P., Advances in Liquid Crystals, G. H. Brown Ed. (Academic Press. New York, 1975).

[2] Bouligand Y., J. Phys. Colloq France 51 (1990) C7-35.

[3] Kelker H. and Hatz R., Handbook of Liquid Crystals

(Verlag

Chemie, Weinheim, Deerfield, 1980).

1060 JOURNAL DE PHYSIQUE II N° 6

[4] Helfrich W., Z. Naiu>fior~ch. 33a (1978j 305.

[5] Leibler S. and Lipowski R.. Phy>. Rev. B 35 (1987) 7004.

[6] Golubovic L. and Lubensky T. C., Phys. Rev. B 39 (1989) 121lo.

[7]

Dimeglio

J.-M.,

Dvolaitzky

M.,

L6ger

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[10] Rogers J. and Winsor P. A., Nature 216 (1967) 477.

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[121 al Nallet F., Laversanne R. and Roux D., J. Phys. ii France 3 (1993j ⁴⁸⁷

b) Roux D. and Nallet F.,

unpublished

neutron

scattering

work _on the quatemary system

investigated in the present paper.

[13] Born M. and Wolf E.,

Principles

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Optics

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[14] Helfrich W. and Servuss R. M., Nuoi>o Cimento 3 (1984) 137.

[15] See for instance Strey R., Schomicker R.. Roux D., Nalletf. and Olsson U., J. Chem. Soc.

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Skouri M., Marignan J.. Appell J. and Porte G., J. Phys. ii France1 (1991) 1121.

[16] de Gennes P.-G. and

Taupin

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[17] Roux D. and

Bellocq

A.-M., Physics of Amphiphiles Micelles. Vesicles and Microemulsions, V.

Degiorgio

and M. Corti Eds. (North-Holland

Physics Publishing,

Amsterdam, 1985)

Bellocq

A.-M.,

Physics

of

Complex

and

Supermolecular

Fluids. S. A. Safran and N. A. Clark Eds.

(John Wiley & Sons, New York, 1987).